When I leave my current job, if I don't notify them of my future
employer's company name and keep quiet on social networks like
LinkedIn etc, can I avoid the risk of them exercising the non compete?
Violating a non-compete agreement and hoping that it simply won't be noticed may reduce the risk but certainly cannot avoid it.
Certainly never exposing your employer's name on any social networks or LinkedIn could help.
But you might also need to avoid any professional networking, avoid conferences, make sure you never encounter colleagues who used to work with you, make sure your spouse, children and friends don't talk about your employer, etc.
You might also want to be sure that your future clients don't know of your prior employer. Many folks who jump to a competitor are expected to bring their former clients along with them. Doing so will clearly increase your risk of being caught and perhaps sued.
You can certainly reduce the risk that your employer finds out about your violation, but you simply cannot avoid it. It's a small world, and some risk will clearly remain.
If you are actually concerned about that, you should consult your lawyer.
I have a non compete at my current job and I notified my future
employer of this.
So your future employer is willing to hire you in spite of your disclosed non-compete agreement. For me, that might say something about your future employer.
Perhaps they have concluded that your non-compete is not enforceable (in which case you might be unnecessarily worried). Or perhaps they just don't care either way (in which case that would worry me).